A simple way to avoid metastable configurations in the density-matrix renormalization-group algorithms

Abstract

We use the spin-1 Heisenberg chain with periodic boundary conditions to ilustrate that the systems get stuck in metastable configurations only when the density-matrix renormalization-group algorithm start with small number of states m. We also show that the convergence of the energies have a huge improvement if we start the algorithm with a large number of states m. Keywords: spin model, DMRG, The density-matrix renormalization-group [1, 2] (DMRG) is one of the most appropriate techniques to study static properties of the one-dimensional systems at zero temperature (for a review see, for example, Refs. 3 and 4). It is also possible to calculate dynamic properties [3][4][5][6][7] and work at finite-temperature through the DMRG [8][9][10]. The main advantage of DMRG, compared with the Lanczos exact diagonalization [11], is its capability to obtain the ground-state properties of very large systems in a well controlled way. Note that it is also possible to investigate large systems by Monte Carlo methods. However, the Monte Carlo technique is not appropriated to study frustrated/fermionic systems due to the "sign" problem.Although the DMRG algorithm was developed for onedimensional systems, it has been used to treat twodimensional systems [12][13][14][15][16][17]. The procedure consists in mapping the low-dimensional model on an one-dimensional model with long range interactions. As first point out by Liang and Pang [12], the energies of two-dimensional systems, converge slower than the ones of one-dimensional systems with short range interactions. The number of states needed to keep a fixed accuracy seems to increase exponentially with the width of the system. A similar effect also appears when we study onedimensional systems with periodic boundary condition (PBC) [1,2]. Since the DMRG was developed, it was observed that the ground state energy (as a function of the number of states m kept in the truncation process) converge faster for the system with open boundary condition (OBC) than the one with PBC. Although it is not completely understood, it seems that everytime that an operator that acts in the left block is directly connected with an operator that acts in the right block (see Fig. 1), the ground state energy convergence is slower. This has been observed for one-dimensional systems as well for the two-dimensional systems.Another difficulty also appears when the left and right blocks are directly connected. Some times, in the simulations, the system gets stuck in some local minimum of energy (see Fig. 3(a)), even working with large values of m [18,19]. This is a serious problem. If the energies do not change increasing m, we may think naively that the true ground state energy was reached. But in fact, the energy found is far from the true ground state. decreasing the computation time as well as the memory used, and more important, (ii) to avoid that the system gets stuck in metastable configurations. As discussed by White [19], the main reason that the simulations get stuck in metastable configurations is due...

N-leg spin-SHeisenberg ladders: A density-matrix renormalization group study

“…In order to avoid metastable configurations we start the truncation process with large values of m (typically we start with m 0 ∼ 1200) [36].…”

The N-Leg spin-S Heisenberg ladders: A DMRG study

“…In order to avoid metastable configurations we start the truncation process with large values of m (typically we start with m 0 ∼ 1200). 36 In the next section we present our results for the spin-S N -leg Heisenberg ladders: estimates of the ground state energy per site (e N ∞ ) as well the estimates of the spin gap (∆ s ) in the thermodynamic limit. We also show estimates of the ground state energy per site of the twodimensional spin-S Heisenberg model.…”

Propriedades estáticas e dinâmicas de sistemas fortemente correlacionados

“…O peso descartado nas varreduras finais foi tipicamente 10-8 -10-12. Para evitar configurações metaestáveis, iniciamos o processo de truncamento com valores grandes de m (tipicamente iniciamos com mo ~ 1200) [127].…”

“…É importante ressaltar que consideramos aqui que uma iteração é composta pelo conjunto da primeira e da segunda parte da varredura. O número de estados retidos é aumentado progressivamente a cada iteração,11 veja por exemplo a referência [127]. A seguir apresentaremos um algoritmo do DMRG para sistema finito.…”